{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib as mp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       " 2.99]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = [float(i)/100.0 for i in range(1,300)]\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = [math.log(i) for i in x]\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x88573c8>]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xa29ac50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'r-', linewidth = 3, label = 'log curve')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = [x[20], x[175]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = [y[20], y[175]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0xa4fbf28>]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xa2cefd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(a, b, 'g-', linewidth=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x88845c0>]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xa4e4ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(a, b, 'b*', markersize=15, alpha=0.75)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "No handles with labels found to put in legend.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0xa2b3c18>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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LUiP+Dw9jnmiFJn0/AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xa5e5e48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.legend(loc='upper left')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x8ec4630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib\n",
    "import math \n",
    "import matplotlib.pyplot as plt \n",
    "\n",
    "plt.plot(x, y, 'r-', linewidth = 3, label = 'log curve')\n",
    "a = [x[20], x[175]]\n",
    "b = [y[20], y[175]]\n",
    "plt.plot(a, b, 'g-', linewidth=2)\n",
    "plt.plot(a, b, 'b*', markersize=15, alpha=0.75)\n",
    "plt.legend(loc='upper left')\n",
    "plt.grid(True)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('log(x)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show>"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x8f99160>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy\n",
    "u = numpy.random.uniform(0.0, 1.0, 10000)\n",
    "plt.hist(u, 80, facecolor='r',alpha=0.75)\n",
    "plt.grid(True)\n",
    "plt.show"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10000\n",
      "10000\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x8e2d400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "times = 10000\n",
    "for time in range(times):\n",
    "    u += numpy.random.uniform(0.0, 1.0, 10000)\n",
    "print(len(u))\n",
    "u /= times\n",
    "print(len(u))\n",
    "plt.hist(u, 80, facecolor='r', alpha=0.75)\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xa54b7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "x = np.arange(0.05, 3, 0.05)\n",
    "y1 = [math.log(a, 1.5) for a in x]\n",
    "plt.plot(x, y1, linewidth=2, color='#007500', label='log1.5(x)')\n",
    "plt.plot([1, 1], [y1[0], y1[-1]], 'r--', linewidth=2)\n",
    "y2 = [math.log(a, 2) for a in x]\n",
    "plt.plot(x, y2, linewidth=2, color='#9F35FF', label='log2(x)')\n",
    "y3 = [math.log(a, 3) for a in x]\n",
    "plt.plot(x, y3, linewidth=2, color='#F75000', label='log3(x)')\n",
    "plt.legend(loc='lower right')\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
